Complexity of (arc)-connectivity problems involving arc-reversals or deorientations

By a well known theorem of Robbins, graph G has strongly connected orientation if and only is 2-edge-connected it easy to find, in linear time, either cut edge or strong G. A result Durand de Gevigney shows that for every k≥3 NP-hard decide given k-strong orientation. Thomassen showed one can check polynomial time whether 2-strong This implies digraph D we determine reorient (=reverse) some arcs D=(V,A) obtain D′=(V,A′). naturally leads the question determining minimum number such reverse before resulting 2-strong. In this paper show finding NP-hard. If 2-connected no orientation, may ask how many its edges orient so mixed still Similarly, remains 2-arc-strong. We prove when restricted graphs satisfying suitable connectivity conditions, both these problems are equivalent must double order 4-edge-connected graph. Using this, all three Finally, consider operation deorienting an arc uv meaning replacing by undirected between same vertices. terms properties, adding opposite vu D. ℓ≥3 find deorient ℓ-strong D′.